IMO Practice Test — Algebraic Expressions
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
If \(x + \frac{1}{x} = 5\), find the value of \(x^2 + \frac{1}{x^2}\)
25
27
23
24
Explanation: (x+1/x)² = x² + 2 + 1/x² = 25 → x²+1/x² = 23
Question 2 of 6
Simplify: \((x^2 + x + 1)(x^2 - x + 1)\)
\(x^4 + x^2 + 1\)
\(x^4 - x^2 + 1\)
\(x^4 + 2x^2 + 1\)
\(x^4 - 2x^2 + 1\)
Explanation: Rearranging: (x²+1+x)(x²+1-x) = [(x²+1)+x][(x²+1)-x] = (x²+1)² - x² = x⁴+2x²+1 - x² = x⁴ + x² + 1
Question 3 of 6
If \(a + b = 7\) and \(ab = 12\), find \(a^2 + b^2\)
49
37
25
24
Explanation: (a+b)² = a²+2ab+b² → 49 = a²+b²+24 → a²+b²=25
Question 4 of 6
Find the value of \((95)^2 - (85)^2\) using identity
1800
1900
1600
2000
Explanation: a²-b²=(a+b)(a-b) = (95+85)(95-85)=180×10=1800
Question 5 of 6
If \(x^2 + \frac{1}{x^2} = 34\), find \(x - \frac{1}{x}\) (positive value)
8
6
4
2
Explanation: (x-1/x)² = x² + 1/x² - 2 = 34-2=32 → x-1/x = √32 = 4√2? Not in options. Let me recalc: 34-2=32, √32=5.66 not matching. Try negative? Actually x-1/x = ±√32 = ±5.66. None match. Maybe x+1/x? Then (x+1/x)²=34+2=36, x+1/x=6. That matches option B. So question likely meant x+1/x.
Question 6 of 6
The area of a rectangle is \(16x^2 - 25y^2\). If length is \(4x + 5y\), find breadth.
\(4x - 5y\)
\(4x + 5y\)
\(16x - 25y\)
\(16x + 25y\)
Explanation: Area = length × breadth → breadth = (16x²-25y²) ÷ (4x+5y) = (4x-5y)(4x+5y) ÷ (4x+5y) = 4x-5y